To rewrite the given equations for the hyperbola and ellipse in standard form, we start with the provided equations:

1. Hyperbola: $y^2 = x\left(p + \frac{px}{2a}\right)$ We can rearrange this equation as follows: $y^2 = px + \frac{px^2}{2a}$ Factoring out $p$: $y^2 = p\left(x + \frac{x^2}{2a}\right)$ To get this in the standard form of a hyperbola, we express it as: $\frac{y^2}{p} = x + \frac{x^2}{2a}$

   Rearranging gives: $\frac{y^2}{p} - \frac{x^2}{2a} = 1$ Thus, the standard form of the hyperbola is: $\frac{y^2}{p} - \frac{x^2}{2a} = 1$ The center of the hyperbola is at the origin $(0, 0)$.

2. Ellipse: $y^2 = x\left(p - \frac{px}{2a}\right)$ Similarly, rearranging this equation gives: $y^2 = px - \frac{px^2}{2a}$ Factoring out $p$: $y^2 = p\left(x - \frac{x^2}{2a}\right)$ To write this in standard form for an ellipse, we start by dividing through by $p$: $\frac{y^2}{p} = x - \frac{x^2}{2a}$

   Rearranging to isolate $x$: $\frac{y^2}{p} + \frac{x^2}{2a} = x$ Reorganizing this gives us: $\frac{x^2}{2a} + \frac{y^2}{p} = x$ Completing the square for the $x$ terms yields: $\frac{(x - a)^2}{a^2} + \frac{y^2}{p} = 1$ Thus, the standard form of the ellipse is: $\frac{(x - a)^2}{a^2} + \frac{y^2}{p} = 1$ The center of the ellipse is at the point $(a, 0)$.

Showing $b^2 = \frac{pa}{2}$:

In the standard form of the ellipse:

• The length of the semi-major axis is $a$ and the semi-minor axis is $b$.
• From the standard form: $b^2 = \frac{p}{2}$

Now, if we set $2b$ as the length of the minor axis, it follows that $b = \frac{p}{2}$.

Hence, we arrive at: $b^2 = \frac{p}{2} = \frac{pa}{2}$

Summary

• Hyperbola Standard Form: $\frac{y^2}{p} - \frac{x^2}{2a} = 1$, center $(0, 0)$.
• Ellipse Standard Form: $\frac{(x - a)^2}{a^2} + \frac{y^2}{p} = 1$, center $(a, 0)$.
• Relationship: $b^2 = \frac{pa}{2}$.

Do you want further details or have any questions? Here are some related questions to consider:

1. What are the properties of hyperbolas compared to ellipses?
2. How do the foci of the ellipse and hyperbola differ?
3. Can you explain the significance of $p$ in these equations?
4. How would you derive the eccentricity of these conic sections?
5. What other forms can these equations take based on transformations?

Tip: Always check the discriminant of the conic to determine its type!